Nature of stationary points. Hello friends, today it’s about the nature of stationary points of any function. Have a look!!

**Nature of stationary points**

Suppose I have a function . And I want to find out the nature of stationary point or points of this function.

**Method **

So I’ll start with the stationary point(s).

**Step 1**

First of all, I’ll differentiate the function partially with respect to to get the value of . Then I’ll differentiate partially with respect to to get the value of . Next, I’ll equate these values to .

If interested, you can also check out: First-order partial derivative of functions with two variables

So I’ll solve and to get the stationary value(s) of the function.

Next, I’ll determine the nature of the stationary points.

**Step 2**

Now I’ll differentiate partially with respect to to get the value of .

Also, I’ll differentiate partially with respect to to get the value of .

And, then I’ll differentiate partially with respect to to get the value of .

If interested, you can also check out: Second-order partial derivative of functions with two variables

Ok, now suppose is a stationary point of the function . And I have to find out its nature.

Then I’ll get the values of and at the point .

Now comes a relation. And this relation is very important to know the nature of the stationary point. So it says

Now if , then I have to check further for maximum or minimum values.

So it looks like

Ok, now I’ll give some examples on how to find out the nature of stationary points of a function.

**The nature of stationary points of any function – Some solved examples**

Note: None of these examples is mine. I have chosen these from some books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2011), “Locate the stationary points of the following function. Determine the nature of the points.

”

**Solution**

Now here the given function is

So I’ll start with the stationary point(s).

**Step 1**

First of all, I’ll differentiate the function partially with respect to to get the value of as

(1)

Next, I’ll differentiate partially with respect to to get the value of as

(2)

Now I’ll get the stationary point of the function . So, I’ll equate the values of and to .

Then I can rewrite equation (1) as

Similarly, I can rewrite equation (2) as

Hence these equations give me two different equations such as

(3)

(4)

Now I’ll solve equations (3) and (4) to get the values of and .

**Step 2**

As I can see from equation (3), I can write as

Then I’ll substitute this value of in equation (4) to get

So this means

Now I’ll simplify it further to get

Thus I can say that

Then the value of will be

Thus I can say that

So the stationary point of the function is

Next, I’ll find the nature of this stationary point.

So, for that, I’ll start with the partial differentiation once more.

**Step 3**

Now I’ll differentiate in equation (1) partially with respect to to get the value of as

(5)

Next, I’ll differentiate in equation (2) partially with respect to to get the value of as

(6)

And, then I’ll differentiate in equation (1) partially with respect to to get the value of as

(7)

Next, I’ll find out the nature of the stationary point using the standard relations.

**Step 4**

As I can see from equations (5), (6) and (7), the values of and are constant. So these values are not dependent on the values of and .

Therefore I can find out the value of from equations (5), (6) and (7) as

So this gives

Now as per the standard formula, if the value of is greater than , then the nature of the stationary point is either maximum or minimum.

Hence in this case also, the nature of the stationary point is either maximum or minimum.

Now I’ll do the next test. And that is, to check the values of and at the stationary point.

As I can see from equation (5), the value of is , that is less than .

Also, I can see from equation (6) that the value of is . So that is also less than .

Since both and are negative, I can say that the function has a maximum value at .

Hence I can conclude that this is the answer to the given example.

Now I’ll give another example.

**Example 2**

According to Stroud and Booth (2011), “Locate stationary points of the function

and determine their nature.”

**Solution**

Now here the given function is

So I’ll start with the stationary point(s).

**Step 1**

First of all, I’ll differentiate the function partially with respect to to get the value of as

(8)

Next, I’ll differentiate partially with respect to to get the value of as

(9)

Now I’ll get the stationary point of the function . So, I’ll equate the values of and to .

Then I can rewrite equation (8) as

Similarly, I can rewrite equation (9) as

Hence these equations give me two different equations such as

(10)

(11)

Now I’ll solve equations (10) and (11) to get the values of and .

**Step 2**

At first, I’ll subtract equation (10) from the equation (11) to get

So this gives

(12)

Therefore I can say, either equals or equals .

If , then .

Now I’ll substitute in equation (10) to get

Then I’ll simplify it to get

Next, I’ll factorise to get

Thus I can say that either or or .

Since , the corresponding values of will be or or .

So I can say that I have got three stationary points such as and .

Now, equation (12) has another factor . At the next step, I’ll work on it.

**Step 3**

So for equals , equals . And this gives .

Next, I’ll substitute in equation (10) to get

Then I’ll simplify it a bit more to get

So this means for , .

If , then the points will be imaginary. So I don’t need to consider this option.

Thus the conclusion is there are three stationary points of the function . And these are

Next, I’ll test the nature of the stationary points. And, for that I’ll start with the second-order partial differentiation.

**Step 4**

Now I’ll differentiate in equation (8) partially with respect to to get the value of as

(13)

Next, I’ll differentiate in equation (9) partially with respect to to get the value of as

(14)

And, then I’ll differentiate in equation (8) partially with respect to to get the value of as

(15)

Next, I’ll find out the nature of the stationary point using the standard relations. And I’ll start with the point .

**Step 5**

First of all, I’ll find out the values of and at the point .

So I’ll substitute at equation (13) to get

And this gives .

Similarly, I’ll substitute at equation (14) to get

And this gives .

Next, I substitute at equation (14) to get

And this gives .

Now I’ll check the relation .

So this gives

Thus at the point , the value of is negative. So this means the point

Now I’ll check the nature of the point .

**Step 6**

Thus I’ll find out the values of and at the point .

So I’ll substitute at equation (13) to get

And this gives .

Similarly, I’ll substitute at equation (14) to get

And this gives .

Next, I substitute at equation (14) to get

And this gives .

Now I’ll check the relation .

So this gives

Thus at the point , the value of is negative. So this means the point

Now I’ll check the nature of the point .

**Step 7**

Thus I’ll find out the values of and at the point .

So I’ll substitute at equation (13) to get

And this gives .

Similarly, I’ll substitute at equation (14) to get

And this gives .

Next, I substitute at equation (14) to get

And this gives .

Now I’ll check the relation .

So this gives

Thus at the point , the value of is negative. So this means the point

Dear friends, this is the end of my today’s post on how to identify the nature of stationary points of any function. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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